Here is the full list of all 52 Summon Night 6: Lost Borders trophies - there are 40 bronze, 9 silver, 2 gold and 1 platinum.
Acquired all trophies
Generated a Summon Assist using 8 people
Cleared all events and battles at S rank
Achieved all Brave Orders
Completed all Night Conversations
Welcomed Karma End
Completed [End of the Illusion, and the Beginning]
Acquired all Summonite leaves
Completed all Town Quests at the Eucross
Generated all types of Summon Bursts
Acquired all Summon Beasts
Created all possible dishes
Upgraded a weapon to the maximum
Completed [The Chosen Future]
Completed all of a character's Night Conversations
Completed [End of Infancy]
Completed [The Line Between Reason and Savagery]
Completed [Truth to Be Mocked]
Completed [From the Cage of Despair]
Completed [Get Those Days Back!]
Completed [Demon of the Mountain Pass]
Completed [The Price Called Bonds]
Completed [The Violent Man and the Runaway Daughter]
Completed [Lamp that Illuminates the Darkness]
Completed [Clash of the Kings]
Upgraded a Summon Beast to the maximum
Completed [Silhouette of Thanatos]
Completed all Fishing Orders
Fished the Cove King
Cleared the lowest level of the Endless Domain
Acquired all of a character's skills
Completed [By the Side of One Yet Unknown]
Used Raj, Amu or Ist's special skills a total of 30 times
Generated Chain Attacks 30 times
Completed [An Administrator's Choice]
Completed [World Cobbled Together]
Completed [An Encounter of Past and Future]
Completed [Brutal Sword and Magical Sword]
Completed [Thou Shalt Not Deceive]
Completed [Raving Madness]
Completed [So Long, Tranquility]
Used Summon Burst for the first time
Completed [Troublesome Girl]
Completed [Prisoners of Delusion]
Defeated 400 enemies
Completed [Demon Lord Attack!]
Completed [Obstinate Sword]
Defeated 100 enemies
Completed [The Pirate, the Angel and the Lost]
Completed [Heroes Falling From the Sky]
Won the first battle
Achieved a Brave Order for the first time